invoke an external web services when an event occurs

I am new to web services. My institution is collaborating with other institution offering their courses on our moodle. They require us to send the user details using their RESFul web services API when the user is enrolled in the courses they offer.

I kinda know the workflow as follows but I don't if I should build a enrol plugin or add to the core code.

1.event:user enrol into courses

2.event handler:invoke the web service to POST the user info to their database.

I am not sure where I should define the event handler

Any help or hint would be much appreciated. If you are based in Melbourne. I am more than happy to shout you coffee or something.